In idempotent analysis, the tropical semiring is a semiring of extended real numbers with the operations of minimum (or maximum) and addition replacing the usual ("classical") operations of addition and multiplication, respectively.

The tropical semiring has various applications (see tropical analysis), and forms the basis of tropical geometry. The name tropical is a reference to the Hungarian-born computer scientist Imre Simon, so named because he lived and worked in Brazil.

Definition

The (or or ) is the semiring (or semifield), with the operations:

are referred to as tropical addition and tropical multiplication is 0.

Similarly, the (or or or ), with operations:

is 0.

, and generally one of these is chosen and referred to simply as the tropical semiring. Conventions differ between authors and subfields: some use the min convention, some use the max convention.

The two tropical semirings are the limit ("tropicalization", "dequantization") of the log semiring as the base goes to infinity (max-plus semiring) or to zero (min-plus semiring).

Tropical addition is idempotent, thus a tropical semiring is an example of an idempotent semiring.

A tropical semiring is also referred to as a , though this should not be confused with an associative algebra over a tropical semiring.

Tropical exponentiation is defined in the usual way as iterated tropical products.

Valued fields

The tropical semiring operations model how valuations behave under addition and multiplication in a valued field is a field equipped with a function

:

Therefore the valuation v is almost a semiring homomorphism from K to the tropical semiring, except that the homomorphism property can fail when two elements with the same valuation are added together.

Some common valued fields:

• ,

• with the p-adic valuation,

• a non-Archimedean local field, such as the p-adic numbers,

• the field of formal Laurent series (integer powers), or the field of Puiseux series, or the field of Hahn series appearing in the series.

References